Control and Synchronization of Neuron Ensembles Jr-Shin Li, Member, IEEE, Isuru Dasanayake, Student Member, IEEE, and Justin Ruths

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 8, AUGUST 2013 1919 Control and Synchronization of Neuron Ensembles Jr-Shin Li, Member, IEEE, Isuru Dasanayake, Student Member, IEEE, and Justin Ruths

Abstract—Synchronization of oscillations is a phenomenon neuroscience devising minimum-power external stimuli that prevalent in natural, social, and engineering systems. Controlling synchronize a population of coupled or uncoupled neurons, synchronization of oscillating systems is motivated by a wide range or desynchronize a network of pathologically synchronized of applications from surgical treatment of neurological diseases neurons, is imperative for wide-ranging applications from the to the design of neurocomputers. In this paper, we study the control of an ensemble of uncoupled neuron oscillators described design of neurocomputers [10], [11] to neurological treatment by phase models. We examine controllability of such a neuron of Parkinson’s disease and epilepsy [12]–[14]; in biology and ensemble for various phase models and, furthermore, study the chemistry, application of optimal waveforms for the entrain- related optimal control problems. In particular, by employing ment of weakly forced oscillators that maximize the locking Pontryagin’s maximum principle, we analytically derive optimal range or alternatively minimize power for a given frequency controls for spiking single- and two-neuron systems, and analyze entrainment range [8], [15] is paramount to the time-scale the applicability of the latter to an ensemble system. Finally, we present a robust computational method for optimal control of adjustment of the circadian system to light [16] and of the spiking neurons based on pseudospectral approximations. The cardiac system to a pacemaker [17]. methodology developed here is universal to the control of general Mathematical tools are required for describing the com- nonlinear phase oscillators. plex dynamics of oscillating systems in a manner that is Index Terms—Controllability, Lie algebra, optimal control, both tractable and flexible in design. A promising approach pseudospectral methods, spiking neurons. to constructing simplified yet accurate models that capture essential overall system properties is through the use of phase model reduction, in which an oscillating system with a stable I. INTRODUCTION periodic orbit is modeled by an equation in a single variable that represents the phase of oscillation [18], [19]. Phase models ATURAL and engineered systems that consist of en- have been very effectively used in theoretical, numerical, and N sembles of isolated or interacting nonlinear dynamical more recently experimental studies to analyze the collective components have reached levels of complexity that are beyond behavior of networks of oscillators [20]–[23]. Various phase human comprehension. These complex systems often require model-based control theoretic techniques have been proposed an optimal hierarchical organization and dynamical structure, to design external inputs that drive oscillators to behave in a such as synchrony, for normal operation. The synchronization desired way or to form certain synchronization patterns. These of oscillating systems is an important and extensively studied include multilinear feedback control methods for controlling phenomenon in science and engineering [1]. Examples include individual phase relations between coupled oscillators [24] and neural circuitry in the brain [2], sleep cycles and metabolic phase model-based feedback approaches for efficient control chemical reaction systems in biology [3]–[5], semiconductor of synchronization patterns in oscillator assemblies [9], [25]. lasers in physics [6], and vibrating systems in mechanical en- These synchronization engineering methods, though effective, gineering [7]. Such systems, moreover, are often tremendously do not explicitly address optimality in the control design large in scale, which poses serious theoretical and computa- process. More recently, minimum-power periodic controls tional challenges to model, guide, control, or optimize them. that entrain an oscillator with an arbitrary phase response Developing optimal external waveforms or forcing signals that curve(PRC)toadesired forcing frequency have been derived steer complex systems to desired dynamical conditions is of using techniques from calculus of variations [15], [26]. In this fundamental and practical importance [8], [9]. For example, in work, furthermore, an efficient computational procedure was developed for optimal control synthesis employing Fourier series and Chebyshev polynomials. Minimum-power stimuli d amplitude that elicit spikes of a single neuron Manuscript received January 03, 2012; revised August 14, 2012; accepted with limite January 09, 2013. Date of publication March 07, 2013; date of current version oscillator at specified times have also been analytically calcu- July 19, 2013. This work was supported in part by the National Science Foun- lated using Pontryagin’s maximum principle, where possible dation under the Career Award 0747877 and the Air Force Office of Scientific neuron spiking range with respect to the bound of the control Research Young Investigator Award FA9550-10-1-0146. Recommended by As- sociate Editor P. Tabuada. amplitude has been completely characterized [27], [28]. In J.-S. Li and I. Dasanayake are with the Department of Electrical addition, charge-balanced minimum-power controls for spiking and Systems Engineering, Washington University, St. Louis, MO asingle neuron has been thoroughly studied [29], [30]. 63130 USA (e-mail: [email protected]; [email protected]; [email protected]). The objective of this paper is to study the fundamental J. Ruths is with the Engineering Systems and Design Pillar, Singa- limit of how the neuron dynamics, described by phase models, pore University of Technology and Design, 138682 Singapore (e-mail: can be perturbed by the use of an external stimulus that is [email protected]). sufficiently weak. We, in particular, investigate controllability Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. and develop optimal controls for spiking an ensemble of Digital Object Identifier 10.1109/TAC.2013.2250112 neurons. In Section II, we briefly introduce the phase model

0018-9286/$31.00 © 2013 IEEE 1920 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 8, AUGUST 2013 for oscillating systems and examine controllability of an ensemble of uncoupled neurons for various commonly used phase models characterized by different baseline dynamics and phase response functions. Then, in Section III, we formulate the optimal control of spiking neurons as steering problems and derive minimum-power and time-optimal controls for single- and two-neuron systems. Finally, we implement a multidimensional pseudospectral method to solve optimal ensemble control problems for spiking an ensemble of neurons, as is proven possible in Section II. This computational method permits us to further explore various objective functions with tunable parameters, which exchange, for example, performance and energy. The methodology developed in this article is universal to the control of general nonlinear phase oscillators. Fig. 1. Free evolution of a Theta neuron. The baseline current , and hence it spikes periodically with angular frequency and period . II. CONTROL OF NEURON OSCILLATORS similar nonlinear systems. Similar control problems of control- A. Phase Models ling a family of structurally similar bilinear systems have been extensively studied in quantum control [32]–[35]. The dynamics of an oscillator are often described by a set of ordinary differential equations that has a stable periodic orbit. B. Controllability of Neuron Ensembles Consider a time-invariant system with , In this section, we analyze controllability properties of where is the state and is the control, and finite collections of neuron oscillators. We first consider the assume that it has an unforced stable attractive periodic orbit Theta neuron model (Type I neurons) which describes both homeomorphic to a circle, satisfying superthreshold and subthreshold dynamics near a SNIPER , on the periodic orbit for (saddle-node bifurcation of a fixed point on a periodic orbit) . This system of equations can be reduced to bifurcation [36], [37]. asinglefirst-order differential equation, which remains valid 1) Theta Neuron Model: The Theta neuron model is charac- while the state of the full system stays in a neighborhood of its terized by the neuron baseline dynamics, unforced periodic orbit [19]. This reduction allows us to repre- , and the PRC, , namely, sent the dynamics of a weakly forced oscillator by a single phase variable that defines the evolution of the oscillation (2)

(1) where is the neuron baseline current. If ,then for all . Therefore, in the absence of the input the neuron where is the phase variable, and are real-valued functions, fires periodically since the free evolution of this neuron system, and is the external stimulus (control) [19], [31]. The i.e., , has a periodic orbit function represents the system’s baseline dynamics and is known as the PRC, which describes the infinitesimal sensitivity for (3) of the phase to an external control input. One complete oscillation of the system corresponds to . In the case of neural oscillators, represents an external current stimulus and with the period and hence the frequency , is referred to as the instantaneous oscillation frequency in the where is a constant depending on the initial condition. For absence of any external input, i.e., . As a convention, a example, if ,then . Fig. 1 shows the free evolution neuron is said to spike or fire at time following a spike at of a Theta neuron with . This neuron spikes periodically time 0 if evolves from to ,i.e.,spikes at with angular frequency .When occur at ,where .Intheabsenceof , then the model is excitable, namely, spikes can occur any input the neuron spikes periodically at its natural fre- with an appropriate input . However, no spikes occur in the quency, while by an appropriate choice of the spiking time absence of an input as can be advanced or delayed in a desired manner. In this paper, we study various phase models characterized by for different and functions in (1). In particular, we investigate the neural inputs that elicit desired spikes for an ensemble of and there are two fixed points (one of which is stable) for isolated neurons with different natural dynamics, e.g., different . oscillation frequencies. Fundamental questions on the control- Now we consider spiking a finite collection of neurons with lability of these neuron systems and the design of optimal inputs distinct natural oscillation frequencies and with positive base- that cause them to spike arise naturally and will be discussed. line currents. This gives rise to a steering problem of the fi- These are related to the control of a collection of structurally nite-dimensional single-input nonlinear control system, LI et al.: CONTROL AND SYNCHRONIZATION OF NEURON ENSEMBLES 1921

,where , , where and the vector ﬁelds and . In the vector form, this system appears as are deﬁned by

. . . . . (4) . . . . .

in which , ,and for all . The system as in (6) is controllable. in which , ,and Proof: It is sufficient to consider the case where for all .Notethat for and , since otherwise they present the same for all . The ultimate proof of our understanding of neural neuron system. Because for all , the free evolution, systems is reflected in our ability to control them; hence, a com- i.e., , of each is periodic for every initial condition plete investigation of the controllability of oscillator popula- , as shown in (3), with the angular frequency tions is of fundamental importance. We now analyze controlla- and the period . Therefore, the free evolution bility for the system as in (4), which determines whether spiking of is periodic with a least period or is recurrent (see Remark or synchronization of an oscillator ensemble by the use of an ex- 1). We may then apply Theorem 1 in computing the reachable ternal stimulus is possible. set of this system. Let denote the Lie Because the free evolution of each neuron system , , bracket of the vector fields and , both definedonanopen in (4) is periodic as shown in (3), the drift term causes no subset of . Then, the recursive operation is denoted as difficulty in analyzing controllability. The following theorem provides essential machinery for controllability analysis. Theorem 1: Consider the nonlinear control system for any ,setting . The Lie brackets of (5) and include

Suppose that and are vector ﬁelds on a manifold .Sup- pose that meet either of the conditions of Chow’s theorem . (described in Appendix A), and suppose that for each initial con- . dition the solution of

. . is periodic with a least period .Thenthe reachable set from for (5) is ,where denotes the Lie algebra generated by the vector fields for , positive integers. Thus, , ,spans and ,and is the smallest subgroup of the at all since and are distinct for diffeomorphism group, , which contains for all . That is, every point in can be reached from any initial [38]. condition ; hence, the system (6) is controllable. Note Proof. (Sketch): The underlying idea of this theorem for that if , , ,spans . dealing with the drift is the utilization of periodic motions along Remark 1: If there exist integers and such that the the drift vector field, , to produce negative drift by for- periods of neuron oscillators are related by for ward evolutions for long enough time. That is to say, if we are all pairs, , then the free evolution of is pe- at and we want to pass backwards along the vector field riodic with a least period. If, however, such a rational number ,wesimplylet be zero and allow the free periodic mo- relation does not hold between any two periods, e.g., tion to bring nearly back to along the integral curve of and , it is easy to see that the free evolution of is al- . If the least period of the periodic solution through most-periodic [39] because the free evolution of each , , is , then, by following for units of time, is periodic. Hence, the recurrence of in (6) together with the we have the same effect as following for units of Lie algebra rank condition (LARC) described above guarantee time. Thus we can, given enough time, reach any point in the the controllability [40]. set from . More details of the proof can Controllability properties for other commonly used phase be found in [38] and the reference therein. models used to describe the dynamics of neuron or other, e.g., Having this result, we are now able to investigate controlla- chemical, oscillators can be shown in the same fashion. bility of a neuron oscillator assembly. 2) SNIPER PRC: The SNIPER phase model is characterized Theorem 2: Consider the finite-dimensional single-input by , the neuron’s natural oscillation frequency, and the nonlinear control system SNIPER PRC, ,where is a model-dependent constant [36]. In the absence of any external input, the neuron (6) spikes periodically with the period .TheSNIPER 1922 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 8, AUGUST 2013

PRC is derived for neuron models near a SNIPER bifurcation where , ; , denoting the which is found for Type I neurons [37] like the Hindmarsh–Rose terminal cost, , denoting the running cost, model [41]. Note that the SNIPER PRC can be viewed as a spe- and are Lipschitz continuous (over the re- cial case of the Theta neuron PRC for the baseline current . spective domains) with respect to their arguments. For spiking This can be seen through a bijective coordinate transformation a neuronal population, for example, the goal is to drive the , ,ap- system from the initial state, ,toafinal state plied to (2), which yields , ,where , . i.e., the SNIPER PRC with . The spiking property, Steering problems of this kind have been well studied, for ex- namely, and is preserved under ample, in the context of nonholonomic motion planning and the transformation and so is the controllability as analyzed in sub-Riemannian geodesic problems [43], [44]. This class of op- Section II-B1. timal control problems in principle can be approached by the More specifically, consider a finite collection of SNIPER neu- maximum principle, however, in most cases they are analyti- rons with and cally intractable especially when the system is of high dimen- , where convention- sion, e.g., greater than three, and when the control is bounded, ally, for . Similar Lie bracket computations as i.e., . in the proof of Theorem 2 result in, for Note that here we consider the constrained optimal control with a bound on the control amplitude as modeled in (7), so that the optimal controls can be derived with respect to any practi- cally allowable bound . As such, these bounded optimal control solutions can be practical while without violating the theoretical “weak forcing” assumption of the phase model as in (1). In the following, we present analytical optimal controls for single- and two-neuron systems and, furthermore, develop and thus ,since for . a robust computational method for solving challenging op- Therefore, the system of a network of SNIPER neurons is timal control problems of steering a neuron ensemble. Our controllable. numerical method is based on pseudospectral approximations 3) Sinusoidal PRC: In this case, we con- which can be easily extended to consider any topologies of sider and neural networks, e.g., arbitrary frequency distributions and ,where for coupling strengths between neurons, with various types of cost . This type of PRCs with both positive and functional. negative regions can be obtained by periodic orbits near the super critical Hopf bifurcation [19]. This type of bifurcation occurs for Type II neuron models like Fitzhugh–Nagumo A. Minimum-Power Control of a Single Neuron Oscillator model [42]. Controllability of a collection of Sinusoidal Designing minimum-power stimuli to elicit spikes or to miti- neurons can be shown by the same construction, from which gate the pathological synchrony of neuron oscillators is desired to improve the efficacy of surgical treatment of certain neurological diseases, such as Parkinson’s disease and dystonia [14], [45], [46]. Optimal controls for spiking a single neuron oscillator can be derived using the maximum principle. In order to illustrate the idea, we consider spiking a Theta neuron, described in (2), with minimum power. In this case, the cost functional is , and the initial and target states are 0 and and then for , . Therefore, , respectively. We first examine the case when the control is the system is controllable. unbounded. The control Hamiltonian of this optimal control problem is defined by ,where III. OPTIMAL CONTROL OF SPIKING NEURONS is the Lagrange multiplier. The necessary conditions for op- The controllability addressed above guarantees the existence timality yield ,and of an input that drives an ensemble of oscillators between any by . With these conditions, desired phase configurations. Practical applications demand the optimal control problem is then transformed to a boundary minimum-power or time-optimal controls that achieve this value problem, which characterizes the optimal trajectories of goal, which gives rise to an optimal steering problem and . We then can derive the optimal feedback law for spiking a Theta neuron at the specified time by solving the resulting boundary value problem:

s.t.

(7) (8) LI et al.: CONTROL AND SYNCHRONIZATION OF NEURON ENSEMBLES 1923 where , which can be obtained according to

(9)

More details about the derivations can be found in Appendix B. Now consider the case when the control amplitude is lim- ited, namely, , . If the unbounded minimum-power control as in (8) satisﬁes for all , then the amplitude constraint is inactive and obviously the optimal control is given by (8) and (9). However, if for some , then the optimal control is characterized by switching between and the bound (see Appendix B):

(10) where the parameter for in (8) is calculated according to the desired spiking time by

(11)

The detailed derivation of the control is given in Appendix B. Fig. 2 illustrates the optimal controls and the corresponding trajectories for spiking a Theta neuron with natural oscillation frequency ,i.e., , , Fig. 2. (a) Minimum-power controls, , for spiking a Theta neuron with and , at various spiking times that are smaller at various spiking times that are smaller and greater and greater than the natural spiking than the natural spiking time subject to . (b) The resulting optimal phase trajectories following . time with the control amplitude bound . Because the unconstrained minimum-power controls for the cases and , calculated according to where (8), satisfy , there are no switchings in these two cases. (15)

B. Time-Optimal Control of Two Neuron Oscillators Our objective is to drive the two-neuron system from Spiking a neuron in minimum time, subject to a given control the initial state to the desired final state amplitude, can be solved in a straightforward manner. Consider with minimum time, where the phase model of a single neuron as in (1), it is easy to see that . The Hamiltonian for this optimal control for a given control bound , the minimum spiking time is problem is given by achieved by the bang-bang control (16) (12) , where and are the multipliers that correspond to the Lagrangian and the system dynamics, respectively, and which keeps the phase velocity, , at its maximum. The min- denotes a scalar product in the Euclidean space . imum spiking time with respect to the control bound ,de- Proposition 1: The minimum-time control that spikes two noted by , is then given by Theta neurons simultaneously is bang-bang. Proof: The Hamiltonian in (16) is minimized by the control for , (17) (13) for , where the sets and . Time-optimal control of where is the switching function defined by .If spiking two neurons is more involved, which can be formulated there exists no nonzero time interval over which ,then as in (7) with the cost functional and with the optimal control is given by the bang-bang form as in (17), where the control switchings are defined at . We show (14) by contradiction that maintaining is not possible for any 1924 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 8, AUGUST 2013 nonzero time interval. Suppose that for some nonzero and are switching points, the corresponding multipliers vanish time interval, ,thenwehave against the control vector field at those points, i.e.,

(18) (20)

(19) Assuming that the coordinate of , our goal is to calculate the switching time, ,intermsof and .In order to achieve this, we need to compute what the relation where denotes the Lie bracket of the vector fields and implies at time 0. This can be obtained by . According to (18) and (19), is perpendicular to both vectors moving the vector along the -trajectory backward from and ,where to through the pushforward of the solution of the variational equation along the -trajectory with the terminal condition at time . We denote by the value of the -trajectory at time that starts at the point at time 0 and by the backward evolution under the variational equa- Since by the non-triviality condition of the maximum tion. Then we have principle, and are linearly dependent on . One can easily show that these two vectors are linearly dependent either when and , and , or and ,where .Thesethreefami- lies of lines represent the possible paths in the state-space where Since the “adjoint equation” of the maximum principle is pre- can be vanished for some non-trivial time-interval. Now we cisely the adjoint equation to the variational equation, it follows show that these are not feasible phase trajectories that can be that the function is constant along the -tra- generated by a control. Suppose that jectory. Therefore, also implies that for some and for some ,where .Wethen have , irrespective of any control input. Hence, (21) the system is immediately deviated from the line .The same reasoning can be used for showing the case of . Since , we know from (20) and (21) that the two vec- Similarly, if for some tors and are linearly dependent. It fol- and for some , in order for the system to remain on the line lows that , it requires that for . However, this occurs only when and (22) ,where ,since . Furthermore, staying on these points is impossible with any con- where is a constant. We make use of a well-known trol inputs since for and ,the Campbell–Baker–Hausdorff formula [49] to expand phase velocities are , which immediately ,thatis, forces the system to be away from these points. Therefore, the system cannot be driven along the path .This analysis concludes that and do not hold simultaneously over a non-trivial time interval. Now, we construct the bang-bang structure for time-optimal A straightforward computation of Lie brackets gives control of this two-neuron system and, without loss of generality, let . Definition 1: We denote the vector fields corresponding to the constant bang controls and by and , respectively, and call the respective trajectories corresponding to them as -and - trajectories. A where , and further- concatenation of an -trajectoryfollowedbya -trajectory is more denoted by , while the concatenation in the reverse order is denoted by . Due to the bang-bang nature of the time-optimal control for this system, it is sufficient for us to calculate the time between consecutive switches, and then the first switching time can be Consequently, we have determined by the end point constraint. The inter-switching time can be calculated following the procedure described in [47], [48]. Let and be consecutive switching points, and let be a -trajectory. Without loss of generality, we assume that this trajectory passes through at time 0 and is at at time .Since LI et al.: CONTROL AND SYNCHRONIZATION OF NEURON ENSEMBLES 1925 which is further simplified to

where

for , 2. This together with (22) yields

(23)

This equation characterizes the inter-switching along the -trajectory, that is, the next switching time can be calculated given the system starting with evolving along the -trajectory. Similarly, the inter-switching along the -trajectory can be calculated by substituting with in (23). Note that the solution to (23) is not unique, and some of the solutions may not be optimal, which can be discarded in a sys- tematic way. The idea is to identify those possible switching points calculated from (23) with that also having the appropriate sign for . We focus on the case where and are linearly independent, since the case for those being linearly dependent restricts the state space to be the curve

Fig. 3. (a) Time optimal control for two Theta neuron system with and to If and are linearly independent, then can be written reach ( ) with the control bounded by and (b) corresponding trajectories. The gray and white regions represent where is negative and as ,where positive, respectively.

Now in order to synthesize a time-optimal control, it remains to compute the first switching time and switching point, since the consequent switching sequence can be constructed there- after based on the procedure described above. Given an initial state ,thefirst switching time and point will be determined according to the target state, e.g., ,where ,insuchawaythatthe As a result, we can write optimal trajectory follows a bang-bang control derived based . Since we know that at switching points on will reach . Under this construction, we may end up , the Hamiltonian, as in (16), and the choice of with a finite number of feasible trajectories starting with either makes . Therefore, at these points, we have -or -trajectory, which reach the desired terminal state. The , and the type of switching can be determined according to minimum time trajectory is then selected among them. the sign of the function .If ,thenitisan to Fig. 3 illustrates an example of driving two Theta neurons switch since and hence changes its sign from positive time-optimally from to with the control bound to negative passing through the switching point, which corre- , where the natural frequencies of the oscillators are sponds to switch the control from to as in (17). Sim- and corresponding ilarly, if ,thenitisa to switch. Therefore, the to , and , .Inthis next switching time will be the minimum nonzero solution to example, the time-optimal control has two switches at the (23) that satisfy the above given rule. For example, sup- and and the minimum time is 5.61. pose that the system is following a -trajectory starting with a switching point . The possible inter-switching C. Simultaneous Control of Neuron Ensembles times , ,with The complexity of deriving optimal controls for higher di- can then be calculated according to (23) based on .Thus,the mensional systems, i.e., more than two neurons, grows rapidly, next switching point is , and it makes sense to find out how the control of two neurons , such that , which corre- relates to the control of many. One may wonder whether it is sponds to an to switch. possible to use a (optimal) control that spikes two neurons to 1926 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 8, AUGUST 2013 manipulate an ensemble of neurons whose natural frequencies lie between those of the two nominal systems. Of course, if trajectories of the neurons with different frequencies have no crossings following a common control input, then the control designed for any two neurons guarantees to bound trajectories of all the neurons with their frequencies within the range of these two nominal neurons, whose trajectories can then be thought of as the envelope of these other neuron trajectories. We now show that this is indeed the case. Lemma 1: The trajectories of any two Theta neurons with positive baseline currents following a common control input have no crossing points. Proof: Consider two Theta neurons modeled by Fig. 4. Controls (top) and state trajectories (bottom) of Theta (left) and Sinu- soidal (right) PRC neuron models (for , , in (26)). The Theta model is optimized for ; the Sinusoidal model is optimized for and the model-dependent constant , both with interpolation nodes. The gray states correspond to uncontrolled state trajectories, and provide a comparison for the synchrony improvement provided by the compensating optimized ensemble control. with positive baseline currents, , and assume that ,whichimplies since , . pseudospectral computational method for constructing optimal In the absence of any control input, namely, ,itisobvious spiking or synchronization controls. that for all since . Suppose that for and these two phase trajectories IV. COMPUTATIONAL OPTIMAL CONTROL OF SPIKING meet at time ,i.e., . Then, we have NEURON NETWORKS and the equality holds As we move to consider the synthesis of controls for neuron only when the neurons spike at time , i.e., ensembles, the analytic methods used in the one and two neuron , . As a result, , because case become impractical to use. As a result, developing com- and , and hence there exist no crossings putational methods to derive inputs for ensembles of neurons between the two trajectories and . is of particular practical interest. We solve the optimal control Remark 2: Note that the same result as Lemma 1 holds problem in (7) using a modified pseudospectral method. Global and can be shown in a similar fashion for both Sinusoidal and polynomials provide accurate approximations in such a method SNIPER phase models, described in Sections II-B2 and II-B3, which has shown to be effective in the optimal ensemble con- when the model-dependent constant if .Con- trol of quantum mechanical systems [50]–[53]. Below, we out- sider two Sinusoidal neurons given by , line the main concepts of the pseudospectral method for optimal ,for with and . Suppose control problems and then show how it can be extended to con- that for and that . sider the ensemble case. Then, we have ,which Spectral methods involve the expansion of functions in terms yields since and of orthogonal polynomial basis functions on the domain . Therefore, because (similar to Fourier series expansion), facilitating high accuracy and , and thus there exist no crossings between with relatively few terms [54]. The pseudospectral method is a the two trajectories and . The SNIPER case can be spectral collocation method in which the differential equation shown following the same procedure. describing the state dynamics is enforced at specific nodes. De- This critical observation extremely simplifies the design of external stimuli for spiking a neuron ensemble with different veloped to solve partial differential equations, these methods oscillation frequencies based on the design for two neurons with have been recently adopted to solve optimal control problems the extremal frequencies over this ensemble. We illustrate this [55], [56]. We focus on Legendre pseudospectral methods and important result by designing optimal controls for two Theta and consider the transformed optimal control problem on the time two Sinusoidal neurons employing the Legendre pseudospectral domain . method, which will be presented in Section IV. Fig. 4 shows The fundamental idea of the Legendre pseudospectral the optimized controls and corresponding trajectories for Theta method is to approximate the continuous state and control and Sinusoidal neurons with their frequencies belonging to functions, and ,by order Lagrange inter- and , respectively. The optimal controls are polating polynomials, and , based on the designed based only on the extremal frequencies of these two Legendre–Gauss–Lobatto (LGL) quadrature nodes, which ranges, i.e., 0.9 and 1.1 for the Theta neuron model and 1.0 and are defined by the union of the endpoints, ,andthe 1.1 for the Sinusoidal model. roots of the derivative of the order Legendre polynomial. This design principle greatly reduces the complexity of Note that the nonuniformity in the distribution of the LGL finding controls to spike a large number of neurons. Although nodes and the high density of nodes near the end points are the optimal control for two neurons is in general not optimal for key characteristics of pseudospectral discretizations by which the others, this method produces a good approximate optimal the Runge phenomenon is effectively suppressed [57]. The control. In the next section, we will introduce a multivariate interpolating approximations of the state and control functions, LI et al.: CONTROL AND SYNCHRONIZATION OF NEURON ENSEMBLES 1927

and can be expressed as functions of the Lagrange polynomials, , given by [58]

(24) and . The derivative of at the LGL node , ,isgivenby

Fig. 5. Control (top) of a Sinusoidal PRC neuron model ( )driving ﬁve frequencies, ,tothedesiredtargets where are elements of the constant at the terminal time with differentiation matrix [54]. Finally, the integral cost functional interpolation nodes. This control yields state trajectories (lower left) and spiking in the optimal control problem (7) can be approximated by the sequences (lower right), which correspond to when the state trajectories cross multiples of . A monotonicity constraint ( ) was imposed to prevent Gauss–Lobatto integration rule, and we ultimately convert the backward evolution of the trajectory. Black coloring indicates a controlled state optimal control problem into the following ﬁnite-dimensional trajectory or spike sequence, whereas gray coloring indicates a trajectory or constrained minimization problem: spike sequence without control.

in Remark 3, this scaling provides a tunable parameter that determines the tradeoff between performance and input energy. s.t. (25) Fig. 4 shows the optimized controls and corresponding trajectories for Theta and Sinusoidal neuron models for , , ,and belongs to and respectively. In this optimization, the controls are optimized over the where , , two neuron systems with extremal frequencies, whose trajecto- , is the target state and are the LGL weights ries form an envelope, bounding the trajectories of other fre- given by ,inwhich quencies in between, as described in Section III-C. We are able is the order Legendre polynomial. Solvers for this type to design compensating controls for the entire frequency band of constrained nonlinear programs are readily available and solely by considering these upper and lower bounding frequen- straightforward to implement. cies. The controlled (black) and uncontrolled (gray) state trajec- Remark 3: (Extension to an infinite ensemble of neuron sys- tories clearly show the improvement in simultaneous spiking of tems) The pseudospectral computational method can be readily the ensemble of neurons. While a bound is necessary to provide extended to consider an infinite population of neurons, for in- in practice, the inclusion of the minimum energy term in the cost stance, with the frequency distribution over a closed interval, function serves to regularize the control against high amplitude . In such a case, the parameterized state values. These smooth controls and state trajectories are approx- function can be approximated by a two-dimensional interpo- imated with interpolation nodes. lating polynomial, namely, , based In Fig. 5, we demonstrate the flexibility of the method to drive on the LGL nodes in the time and the frequency domain. multiple Sinusoidal ( ) neurons to different desired tar- Similarly, the dynamics of the state can be expressed as an alge- gets. In particular we seek to simultaneously spike five frequen- braic constraint and a corresponding minimization problem can cies with widely dispersed frequency values at a time different be formed [51]. from their natural period. In this figure we consider the frequencies and design controls to A. Optimized Ensemble Controls drive these systems to , respectively, at a We can now apply the above methodology to synthesize op- time . The control shown in the figure is opti- timal controls for neuron ensembles. Since neurons modeled by mized for minimum energy ( , ) and achieves a the SNIPER PRC are special cases of the Theta neuron, here transfer with terminal error of (here we consider Theta and Sinusoidal neuron models. The compu- we impose the desired terminal state as an additional constraint, tational method outlined above permits a flexible framework to rather than incorporate it into the cost function). We also in- optimize based on a very general cost functional subject to gen- clude the realistic constraint for the state to be monotonically eral constraints. We illustrate this by selecting an objective of increasing ( ), which prohibits backward evolution of the the type neuron. The spike train shows that the controls are able to ad- vance the firing of each neuron so that all spike simultaneously (26) at the desired terminal time. Again the gray coloring indicates uncontrolled trajectories or spike trains and offers a comparison which minimizes the terminal error and input energy with a rel- of improvement in synchrony. ative scaling given by the constants and . In highly complex Similarly, Fig. 6 presents the controls and state trajectories problems, such as those given by ensemble systems described corresponding to the minimum energy transfer for Theta neu- 1928 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 8, AUGUST 2013

3 can be employed to calculate optimal controls for spiking or synchronizing networks of coupled neurons with their natural frequencies varying on a continuum. Because phase models are reductions of original higher dimensional state-space systems, there is a fundamental need to explore the limits of such approximations. Therefore, applying the optimal controls derived based on phase-reduced models to the corresponding full state-space models is an important validation and essential future work.

APPENDIX A Fig. 6. Controls and amplitude constrained controls, , upper left and right respectively, of a Theta PRC neuron model driving five fre- CHOW’S THEOREM quencies, ,tothedesiredtargets Theorem 3: (Versions of Chow’s Theorem) Let at the terminal time with interpolation nodes. These controls yield highly similar state trajec- be a collection of vector fields tories (lower left, shown for the unconstrained control) and spiking sequences such that the collection is (lower right, shown for the constrained control), which correspond to when the a) analytic on an analytic manifold . Then given any state trajectories cross multiples of . A monotonicity constraint ( ) was imposed to prevent backward evolution of the trajectory. Black coloring point , there exists a maximal submanifold indicates a controlled state trajectory or spike sequence, whereas gray coloring containing such that indicates a trajectory or spike sequence without control. . b) on a manifold with dim ( ) rons of five frequencies constant on . Then given any point ,there driving them to when .In exists a maximal submanifold containing this case, we demonstrate the capacity to synthesize both un- such that . constrained and constrained ( ) controls. In both cases, the For more details about Chow’s theorem, the reader can refer to state trajectories (unconstrained case) and spike sequence (con- [38], [59], [60]. strained case) show the monotonic state evolution, achieving the desired terminal state within an error ( )of APPENDIX B 6 (unconstrained) and 3 (constrained). OPTIMAL CONTROL OF A SINGLE THETA NEURON Unbounded Minimum-Power Control of a Theta Neuron: The minimum-power control of a single Theta neuron is formu- V. C ONCLUSION lated as In this paper, we considered the control and synchronization of a neuron ensemble described by phase models. We showed that this ensemble system is controllable for various commonly used phase models. We also analytically derived s.t. minimum-power and time-optimal controls for spiking single and two neuron systems. In addition, we adopted a computational pseudospectral method for constructing optimal controls We then can form the control Hamiltonian, that spike neuron ensembles, which demonstrated the underlying controllability properties of such neuron systems. This (27) work characterizes the fundamental limit of how the dynamics of neurons can be perturbed by the use of an external input, where is the Lagrange multiplier. The necessary conditions and the resulting methodology can be applied not only to for optimality from the maximum principle yield neuron oscillators but also to any oscillating systems that can be represented using similar model reduction techniques such (28) as biological, chemical, electrical, and mechanical oscillators. A compelling extension of this work is to consider networks of and . Thus, the optimal control coupled oscillators, whose interactions are characterized by a satisfies coupling function acting between each pair of oscillators. For example, in the well-known Kuramoto’s model, the coupling (29) between the -pair is characterized by the sinusoidal function of the form [18]. The procedure With (29) and (28), this optimal control problem is transformed presented in Theorem 2 can be immediately applied to examine to a boundary value problem, whose solution characterizes the controllability of interacting oscillators by investigating the optimal trajectories recurrence properties [40] of the vector field ,andtheLie algebra . Similarly, the pseudospectral method presented in Section IV and its extension addressed in Remark (30) LI et al.: CONTROL AND SYNCHRONIZATION OF NEURON ENSEMBLES 1929

(31) Note that the number of time intervals that defines the number of switches in the optimal control law. Specif- ically, if for time intervals, then the optimal with boundary conditions and , while controlwillhave switches. Here we consider the simplest ,and are unspecified. case, where the optimal control has only two switches, which is Additionally, since the Hamiltonian is not explicitly depen- actually the only case for the Theta neuron model. As a result, dent on time, the optimal triple satisfies suppose that for only one time interval, and then , ,where is a constant. Together with (29) and there are two switching angles and at which (27), this yields .Since . These two conditions, together with (11), deter- , ,where is undetermined. The optimal mine the unknown parameters , ,and that characterize multiplier can be found by solving the above quadratic equa- the bounded optimal control, , as given in (10) for the speci- tion, which gives fied spiking time . Note that the range of feasible spiking times is determined by the bound of the control amplitude .Acom- plete characterization of possible spiking range can be found in (32) [27]. and then, from (30), the optimal phase trajectory follows

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